1 Multilinear algebra
Tensors. Tensors in physics. The external algebra.
2 Representations of finite groups
General theory. Characters. Irreducible representations.
3 Tensors and symmetric group
Introduction. Schur functors. Examples.
4 Introduction to differential geometry
Differential varieties. Fiber bundles. Differential forms.
5 Lie Groups and Lie algebras
Lie groups. Simple Lie algebras. Representations of simple Lie algebras. Representations of sl(2).
6 Classification of simple Lie algebras
Cartan algebra and weights. Roots and Dynkin diagrams. Real simple algebras.
7 Representations of simple Lie algebras
Weights: integrality and symmetries. The highest weight. Irreducible representations of sl(n). The case n=3.
8 Orthogonal Groups
Bilinear and quadratic forms. Orthogonal groups. Quaternions. The group SO(2m). The group SO(2m+1).
9 Spin representations
Clifford algebras. The spin group. Spin representations of so(n).
10 Geometric structures
Fiber bundles. Geometry and Lie groups. Connections. Covariant derivatives. Curvature. Riemannian structure.
11 Dynamical theory of symmetries
Matter fields. Gauge fields. Yang-Mills theories. External symmetries, and relativity.
12 Scalar and spinoral fields
Particles and the Poincaré group. Klein-Gordon equation. Dirac equation. The SO(1,3) group. Spin 1 fields. Fields and gravity.
13 Geometry of the Standard Model of particles and GUT
The field content in the Standard Model. Constructing the Standard Model. Spontaneous symmetry breaking. The mass of neutrino. The seesaw mechanism. GUTs.